I tried to prove that every polynomial of the form
$f(m,n) := m\cdot x^{n-m}+(m+1)\cdot x^{n-m-1}+\cdots+(n-1)\cdot x+n \quad \text{with} \quad 0 < m < n$
is irreducible over the rationals for all integers m and n.
I made some progress using the enestroem-kakeya-theorem and could show that $f(m,n)$ never has a linear factor. If I did not make any mistake, I even proved that $f(m,n)$ is irreducible if its degree is 100 or less.
But I did not find a complete proof even for the case $m=1$. Perhaps, someone has an idea.
What could be useful : Any root $u$ of $f(m,n)$ has the property $\frac{n}{n-1} < |u| < \frac{m+1}{m}$